Oxford definitions for them:
stalk:
1 The main stem of a herbaceous plant.
‘he chewed a stalk of grass’
1.1 The slender attachment or support of a leaf, flower, or fruit.
‘the acorns grow on stalks’stem:
The main body or stalk of a plant or shrub, typically rising above ground but occasionally subterranean.
1.1 The stalk supporting a fruit, flower, or leaf, and attaching it to a larger branch, twig, or stalk.
Best Answer
The venerable OED (as well as your reference) uses each term in the other's definition, but they aren't really interchangeable.
For stalk:
For stem:
Botanists and arborists will usually use stem to refer to a slender portion of the plant, while stalk refers to something more substantial, often the main upright "load-bearing" portion of a plant's body. A stalk can bear several stems, but a stem cannot bear a stalk.
In her blog called BotanicalAccuracy.com, Dr. Lena Struwe uses rhubarb to bring into sharp relief the distinction between stem and stalk. Dr. Struwe does such a good job of it that rather than paraphrase it, I reproduce† it here against the possibility that her blog may disappear one day.
†I have taken the liberty of correcting a few disagreements in number, and adding a few changes for clarity in Dr. Struwe's text. She is after all a botanist, and not merely a harmless drudge.